Title: "Unification" of BSM Searches and SM Measurements: the case of lepton$+MET$ and $m_W$

Abstract: We develop the idea that the unprecedented precision in Standard Model (SM) measurements, with further improvement at the HL-LHC, enables new searches for physics Beyond the Standard Model (BSM).As an illustration, we demonstrate that the measured kinematic distributions of the lepton$+MET$ final state not only determine the mass of the $W$ boson, but are also sensitive to light new physics. Such a search for new physics thus requires a simultaneous fit to the BSM and SM parameters, "unifying" searches and measurements at the LHC and Tevatron. In this paper, we complete the program initiated in our earlier work arXiv:2310.13687. In particular, we analyze ($i$) novel decay modes of the $W$ boson with a neutrinophilic invisible scalar or with a heavy neutrino; ($ii$) modified production of $W$ bosons, namely, associated with a hadrophilic invisible $Z^\prime$ gauge boson; and ($iii$) scenarios without an on-shell $W$ boson, such as slepton-sneutrino production in the Minimal Supersymmetric Standard Model (MSSM). Here, we complement our previous MSSM analysis in arXiv:2310.13687 by considering a different kinematic region. Our results highlight that new physics can still be directly discovered at the LHC, including light new physics,via SM precision measurements. Furthermore, we illustrate that such BSM signals are subtle, yet potentially large enough to affect the precision measurements of SM parameters themselves, such as the $W$ boson mass.